nLab
(infinity,1)-Kan extension

under construction

Idea

The notion if $(\mathrm{\infty },1)$-Kan extension is the generalization of the notion of Kan extension from category theory to (∞,1)-category theory.

Definition

General abstract definition

Independent of any models or concrete realizations chosen, the notion of $(\mathrm{\infty },1)$-Kan extension is intrinsically determined from just the notions of

• (∞,1)-functor,

• (∞,1)-category of (∞,1)-functors

• adjoint (∞,1)-functor.

In terms of these, for $f:C\to C\prime$ any (∞,1)-functor and any (∞,1)-category $A$, there is an induced $(\mathrm{\infty },1)$-functor $f^{*}:{\mathrm{Func}}_{(\mathrm{\infty },1)}(C\prime ,A)\to {\mathrm{Func}}_{(\mathrm{\infty },1)}(C,A)$.

The left $(\mathrm{\infty },1)$-Kan extension functor is the left adjoint (∞,1)-functor to $f^{*}$.

The right $(\mathrm{\infty },1)$-Kan extension functor is the right adjoint (∞,1)-functor to $f^{*}$.

Given different incarnations of or models for the notion of (∞,1)-category, there are accordingly different incarnations and models of this general abstract prescription.

In terms of quasi-categories

…

In terms of Kan-complex enriched categories

see homotopy Kan extension

In terms of simplicial model categories

see homotopy Kan extension

Related concepts

• Kan extension

• $(\mathrm{\infty },1)$-Kan extension

References

$(\mathrm{\infty },1)$-Kan extensions in terms of quasi-categories are discussed in section 4.3 of

• Jacob Lurie, Higher Topos Theory .

For simplicially enriched categories and model categories a discussion is in section A.3.3 there.